This is a question that is often asked on ukulele forums, and I thought it would be helpful to give you my perspective on cleaning the Uke.
I will start off my stating that a ukulele, if looked after, really shouldn't need a great deal of cleaning - if it does - what the heck are you doing with it??! That said, if gigging, camping etc, you can get the instrument dirty with grime, sweat etc, and just day to day playing will cause a build up of grease and oil from your hands. Mix this with natural dust and you get a claggy ukulele!
I will also add from the start that the instrument cleaning business is a lucrative market but in my view 95% of the products on offer are overkill. They can be avoided completely in my view if you keep a lint free soft cloth in your ukulele case and EVERY time you finish playing you give the whole instrument a polish all over. Do this and I think you will rarely need any sort of cleaning fluid or polish.
Anyway, if you ARE interested in cleaning and sprucing up your ukulele read on.
Cleaning bodies will depend on how dirty. If you have dried on grime on there, mud, beer, whatever - the best thing to do is to clean off with a barely damp lint free cloth, then polish with a dry clean cloth.
If you have a gloss or painted body uke, you can give a bit of showroom shine using a small (tiny) amount of guitar polish on a soft cloth and polishing according to the instructions. Be careful though - make sure your cloth is super super soft with no debris on it. The sign of an over polished instrument are swirl marks in the varnish. Bit like on an over polished car really. Treat your uke as you would cleaning a Ferrari!
If you are using a polish - keep it well away from the fingerboard and strings!!
If you have a natural finish uke in matte - be very careful. In my youth I learned a lesson with those applying polish which then sunk into the grain and left my instrument spotty! Not good
If you have pearly, plastic or chrome tuning pegs and strap buttons, they really should only need a buff with a soft cloth
3. Fingerboard - the fingerboard, if natural and not painted is a different animal. Do not apply polish to it as you may cause it to expand and then have problems with frets rising or falling out. If you have a uke with a seriously grimy fingerboard between frets, you can VERY carefully and lightly rub between them with a super fine grain wire wool to remove the gunk and get back to the wood. You can think about treating your fingerboard with a bore oil (avoid lemon oil sold in guitar shops - it is synthetic) - I use Fret doctor bore oil which is the choice of oil for woodwind instrument players. Its totally natural. Apply is very very very lightly, let dry and polish off. One of the worst things you can do with a fingerboard is over treat it. The regularity in applying oil depends on your climate but certainly not more than twice a year is needed.
4. Strings - being nylon, strings on the ukulele dont deteriorate as quickly as steel strings on a guitar - that said, its nice to keep them clean as they do pick up oils from your fingers. Simply ensure you wipe your strings down after each session - a good habit to form
And honestly, thats about it. Look after your instrument and you will need to apply only very occasional care to keep it looking nice.
Oh, and keep your uke in a quality case when you are not playing to keep the dust off!